The L(log⁡L)ϵL(log⁡L)ϵ endpoint estimate for maximal singular integral operators

We prove in this paper the following estimate for the maximal operator T⁎T⁎ associated to the singular integral operator T:‖T⁎f‖L1,∞(w)≲1ϵ∫Rn|f(x)|ML(log⁡L)ϵ(w)(x)dx,w≥0,0<ϵ≤1. This follows from the sharp LpLp estimate‖T⁎f‖Lp(w)≲p′(1δ)1/p′‖f‖Lp(ML(log⁡L)p−1+δ(w)),1